WO2022087770A1 - 一种基于非梯度拓扑优化的声学超材料设计方法 - Google Patents
一种基于非梯度拓扑优化的声学超材料设计方法 Download PDFInfo
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- the invention belongs to the field of structure and multidisciplinary optimization design, and relates to a non-gradient topology optimization and design method of acoustic metamaterials. This method is suitable for the topology optimization design of acoustic metamaterials and advanced acoustic components.
- Acoustic metamaterials refer to a new class of artificial periodic materials that can control the propagation of acoustic or elastic waves. Its remarkable feature is that it can block the propagation of acoustic and elastic waves in a specific frequency range, and has broad potential in practical applications.
- topology optimization has been successfully applied in the design of phononic crystals, and some novel and attractive acoustic functional devices have been obtained through topology optimization.
- the existing gradient-based topology optimization methods used in the design of acoustic metamaterials face inherent difficulties, that is, the optimal design of acoustic metamaterials is a typical multi-local solution problem with serious initial solution dependence.
- the initial guess (or initial design) can significantly affect the optimal configuration of the microstructure, get trapped in local solutions, and even make the optimization process difficult to converge.
- the topology optimization design method based on intelligent algorithms such as genetic algorithm has a very limited design space due to its huge computational load, and the optimization effect is not good.
- the present invention proposes a method for describing a few parameters of the unit cell topology of acoustic metamaterials, and establishes an acoustic metamaterial topology optimization model based on a non-gradient optimization algorithm, which can be applied to the full band gap of acoustic metamaterials. and the optimal design problem of the directional band gap.
- This method realizes the topological characterization of the complex acoustic metamaterial unit cell structure and the mapping of material properties through a small number of independent design variables, greatly reduces the computational complexity of the acoustic metamaterial unit cell topology optimization, and can effectively overcome the local maximum in the acoustic metamaterial design.
- the optimal solution is difficult, and it is suitable for the optimal design of multiple functional acoustic metamaterials with full band gaps and directional band gaps.
- the present invention provides an effective non-gradient topology optimization method for acoustic metamaterial band gap microstructures.
- This method can solve the design problem of complex acoustic metamaterial unit cells with a small number of design variables and surrogate model optimization algorithms, without the sensitivity information of optimization objectives and constraint functions, and can be directly extended to metamaterial designs with other complex acoustic properties, which is easy to integrate with.
- Various finite element commercial and self-developed software are connected.
- the invention is suitable for the optimal design of multidisciplinary metamaterial microstructures and sensitive components such as acoustics and optics.
- An acoustic metamaterial design method based on non-gradient topology optimization including the following steps:
- ⁇ (r) represents the material density
- U and represent the displacement and acceleration vectors along the x, y and z directions of the coordinate axes, respectively
- ⁇ (r) and ⁇ (r) represent the Lame coefficients that vary with position
- k represents the plane wave vector
- K(k) represents the plane wave vector related
- M is the overall mass matrix of the periodic unit cell
- ⁇ is the circular frequency.
- step S2 Based on the Floquet–Bloch theorem, periodic boundary conditions are introduced into the generalized eigenvalue equation established in step S1 ( Among them, U k (r) represents the periodic displacement field with the same periodicity as the position vector r); based on the finite element method, the eigenvalue problem of the microstructure is discretized.
- step S3 Characterize and map the material distribution and equivalent material properties in the discrete acoustic metamaterial unit cell in step S2.
- the RAMP material interpolation model is introduced to establish the mapping relationship between the relative density of the element and the equivalent Young's modulus of the material.
- f F ( ⁇ ) is the full-bandgap objective function
- N P is the total number of observation points
- Wi is the characteristic matrix related to the ith observation point
- n k is the calculation point of the boundary of the reduced Brillouin zone number.
- f P ( ⁇ ) is the 0-degree directional bandgap objective function
- g P is the 45-degree directional pass-band constraint function.
- loop is the number of sub-optimization problems
- f obj ( ⁇ ) is the objective function of the sub-optimization problem
- f( ⁇ ) is the objective function of the original optimization problem
- p 0 is the transformation of the constrained optimization problem into an unconstrained optimization problem
- the penalty function multiplier of ; ⁇ k is the feasible region of the sub-optimization problem.
- the adaptive design space adjustment strategy is adopted to limit the degree of change in the single optimization update of the material layout of the acoustic metamaterial unit cell, and the optimization model in step S6 is solved based on the sequential Kriging surrogate model optimization algorithm until the optimization problem converges.
- S31 Use material constant function Describe the layout of the two-phase material in the unit cell of the acoustic metamaterial, and select the observation points uniformly in the unit cell of the metamaterial.
- the beneficial effects of the invention are as follows: the method realizes the topological characterization of the complex acoustic metamaterial unit cell structure and the mapping of the material properties through a small number of independent design variables, greatly reduces the computational complexity of the acoustic metamaterial unit cell topology optimization, and can effectively overcome the
- the local optimal solution in material design is difficult, and it is suitable for the optimal design of various functional acoustic metamaterials such as full band gap and directional band gap.
- Metamaterial optimization design and can be directly extended to other metamaterial designs with complex acoustic properties, which is easy to interface with various finite element commercial and self-developed software
- FIG. 1 is a microstructure design of a full-bandgap and directional bandgap material provided by an embodiment of the present invention.
- Fig. 2 Flow chart of acoustic metamaterial optimization based on non-gradient optimization algorithm
- Fig. 3(a) The optimal topological configuration of a specified order full-bandgap acoustic metamaterial
- Figure 3(b) is the acoustic wave transmission amplitude cloud diagram of the full-bandgap acoustic metamaterial structure at the specified frequency (30kHz);
- Figure 3(c) is the acoustic wave transmission amplitude cloud diagram of the full-bandgap acoustic metamaterial structure at the specified frequency (50kHz);
- Fig. 4(a) The optimal topological configuration of the acoustic metamaterial with a specified order directional bandgap
- Figure 4(b) Amplitude cloud diagram of incident acoustic wave transmission on the left side of the waveguide composed of directional bandgap acoustic metamaterials at the specified frequency (20kHz);
- Figure 4(c) Amplitude cloud diagram of incident acoustic wave transmission on the right side of the waveguide composed of directional bandgap acoustic metamaterials at the specified frequency (20kHz);
- Figure 4(d) Amplitude cloud diagram of incident acoustic wave transmission on the left side of the waveguide composed of directional bandgap acoustic metamaterials at the specified frequency (33 kHz);
- Fig. 4(e) Amplitude cloud diagram of the incident acoustic wave transmission on the right side of the waveguide composed of a directional bandgap acoustic metamaterial at the specified frequency (33 kHz).
- a topology optimization design method for acoustic metamaterials based on a non-gradient optimization algorithm uses a material field series expansion strategy to achieve a small number of parameters to characterize any topology of an acoustic metamaterial unit cell, and establish an interpolation model between these design parameters and material properties. and the mapping relationship of the dynamic properties of acoustic materials.
- the dispersion curve of the acoustic metamaterial is obtained, the topology optimization model of the acoustic metamaterial with full band gap and directional band gap is established, and the optimization problem is solved based on the sequential Kriging surrogate model optimization algorithm.
- the optimization process avoids the sensitivity information for solving complex acoustic metamaterial design problems, and can effectively overcome the trap of local optimal solutions.
- Figure 1 depicts a schematic diagram of the optimization of the microstructure design of full-bandgap and directional bandgap materials.
- the present invention takes a typical lead-epoxy acoustic metamaterial unit cell as an example, and the size of the unit cell is a square with a side length of 0.02m.
- the 6th-7th order full bandgap and the 3rd-4th order 0-degree directional bandgap are selected as the objectives of the optimization problem.
- Figure 2 is a flow chart of the implementation of the method.
- the topology optimization objective function can be expressed as max:
- the material field function bound constraint function ⁇ T W i ⁇ 1, (i 1,2,...,1600) was introduced to make the material characterization function value between Then, the topology optimization problem model is established (the relevant optimization calculation results are shown in Figure 3):
- the adaptive design space adjustment strategy is adopted to limit the change degree of the single optimization update of the material layout of the acoustic metamaterial unit cell, and the optimization problem is solved based on the sequential Kriging surrogate model optimization algorithm until the optimization problem converges.
- Fig. 3(a) The optimal microstructure of acoustic metamaterials obtained by the non-gradient topology optimization method of acoustic metamaterials aiming at the 6-7th order full band gap is shown in Fig. 3(a).
- the optimized macrostructure of the full-bandgap acoustic metamaterial is shown in Fig. 3(b) at 30 kHz and 50 kHz for the acoustic wave transmission amplitude nephogram. Sound waves that are not within its band gap pass through.
- the optimal microstructure of the acoustic metamaterial obtained by the non-gradient topology optimization method of the acoustic metamaterial targeting the 3-4 order directional band gap is shown in Fig. 4(a).
- the wave guide structure composed of directional acoustic metamaterial microstructures is shown in Fig. 4(b) for the acoustic wave transmission amplitude nephogram at 20 kHz and 33 kHz.
- the results show that the directional band gap acoustic metamaterial can block 0 degrees in the band gap frequency range. The sound waves propagating in the direction of 45 degrees can pass freely, and the sound waves in all directions outside the band gap can pass through.
- the invention can effectively overcome the initial solution dependence and local optimal solution trap problems in the topology optimization design of the acoustic metamaterial, and can obtain the optimal configuration of the acoustic metamaterial with various order band gaps stably and efficiently.
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Abstract
一种基于非梯度拓扑优化的声学超材料设计方法,主要包括声学超材料单胞拓扑的少量参数描述、基于非梯度优化算法的声学超材料拓扑优化模型两部分。基于材料级数场展开策略,提出声学超材料单胞拓扑的少量参数描述方法,并建立基于非梯度优化算法的声学超材料拓扑优化模型,可适用于声学超材料的全带隙和方向性带隙的优化设计问题。该方法通过少量独立设计变量,实现复杂声学超材料单胞结构的拓扑表征与材料性能的映射,降低声学超材料单胞拓扑优化计算量,可有效克服声学超材料设计中的局部最优解困难,适用于全带隙、方向性带隙等多种功能声学超材料优化设计问题;且不需要带隙特性的灵敏度信息,可实现任意阶带隙的声学超材料优化设计。
Description
本发明属于结构与多学科优化设计领域,涉及一种声学超材料的非梯度拓扑优化与设计方法。本方法适用于声学超材料及先进声学元器件的拓扑优化设计。
声学超材料是指具有控制声波或弹性波传播的一类新型人造周期性材料,其显著特点是可以阻断声波和弹性学波在特定频率范围内的传播,在实际应用中具有广阔的潜力。拓扑优化作为强大的设计工具,已成功应用于声子晶体的设计中,并且通过拓扑优化获得了一些新颖且有吸引力的声学功能器件。
然而,已有声学超材料设计采用的梯度类拓扑优化方法面临着难以解决的本质困难,即声学超材料优化设计为一典型多局部解问题,存在严重的初始解依赖性。在基于梯度的拓扑优化设计方法中,初始猜测(或初始设计)会显著影响微结构的优化配置,陷入局部解,甚至导致优化过程难以收敛。此外,基于遗传算法等智能类算法的拓扑优化设计方法,因其计算量巨大,使得设计空间十分有限,优化效果欠佳。
本发明基于材料级数场展开策略,提出一种声学超材料单胞拓扑的少量参数描述方法,并建立基于非梯度优化算法的声学超材料拓扑优化模型,可适用于声学超材料的全带隙和方向性带隙的优化设计问题。本方法通过少量独立设计变量,实现了复杂声学超材料单胞结构的拓扑表征与材料性能的映射,大大降低了声学超材料单胞拓扑优化计算量,可有效克服声学超材料设计中的局部最优解困难,适用于全带隙、方向性带隙的多种功能声学超材料优化设计问题。
发明内容
针对传统梯度类拓扑优化方法在求解声学超材料优化过程中存在的严重初始解依赖性和易于陷入局部解等缺点,本发明提供一种有效的声学超材料带隙微结构非梯度拓扑优化方法。该方法可用少量设计变量和代理模型优化算法解决复杂声学超材料单胞的设计问题,不需要优化目标及约束函数的灵敏度信息, 并且可以直接扩展到其他复杂声学特性的超材料设计中,便于与各种有限元商业和自研软件进行对接。本发明适用于多学科的超材料微结构及声学、光学等敏感元器件的优化设计。
为了达到上述目的,本发明采用的技术方案为:
一种基于非梯度拓扑优化的声学超材料设计方法,包括以下步骤:
S1:将时间域声学超材料中弹性波的波传导问题转换为频率域下微结构的弹性波的广义特征值问题(将时间域的波传导方程
转化为频率域的广义特征值方程(K(k)-ω
2M)U=0),采用平面应变单元对声学材料单胞进行离散,得到离散后的微结构总体刚度矩阵和总体质量矩阵。
公式中字母指代为:r={x,y}表示位置向量;ρ(r)表示材料密度;U和
分别表示沿坐标轴x,y和z方向的位移和加速度向量;λ(r)和μ(r)表示随位置变化的拉梅系数;k表示平面波矢;K(k)表示与平面波矢相关的周期单胞总体刚度矩阵;M表示周期单胞的总体质量矩阵;ω表示圆频率。
S2:基于Floquet–Bloch定理,在步骤S1中建立的广义特征值方程中引入周期性边界条件(
其中,U
k(r)表示与位置向量r有相同周期性的周期位移场);基于有限元方法对微结构的特征值问题进行离散。
S3:对步骤S2中离散的声学超材料单胞内材料分布与等效材料属性进行表征与映射。基于材料场级数展开策略,引入材料场函数
和距离相关函数C(r
i,r
j)描述单胞内部材料分布;利用级数展开降维技术,将声学超材料单胞内的材料分布用M个独立设计变量ξ={ξ
1ξ
2…ξ
M}
T进行表征。进而,引入RAMP材料插值模型,建立单元相对密度与材料等效杨氏模量的映射关系。
S4:对步骤S3中声学超材料微结构的广义特征方程求解中,设置不同大小的平面波矢量k的入射波对简约布里渊区边界进行扫略,在简约布里渊区的ГXM边界等距选取0度、45度、90度波矢各5组,通过求解任意可能的波矢k 求解广义特征值问题(K(k)-ω
2M)U=0。计算声学超材料的能带结构,并得到指定阶频率对应的带隙ω
j(k
i),式中,k
i表示选取的第i个代表性平面波矢量,ω
j(k
i)表示在指定平面波矢量k
i下单胞的第j阶特征圆频率。
S5:根据步骤S4中带隙计算结果,以ξ={ξ
1ξ
2…ξ
M}
T为设计变量,建立全带隙或方向性带隙的声学超材料拓扑优化模型:
(1)若指定第j阶和j+1阶的全带隙为目标,建立全带隙拓扑优化问题模型:
s.t.(K(k)-ω
2M)u=0
ξ
TW
iξ≤1,(i=1,2,…,N
P).
式中,f
F(ξ)为全带隙目标函数;N
P为总观察点个数;W
i为与第i个观察点相关的特征矩阵;n
k为简约布里渊区边界的计算点个数。
(2)若指定第j阶和j+1阶的方向性带隙为目标,建立局部带隙拓扑优化问题模型:
s.t.(K(k)-ω2M)u=0
ξ
TW
iξ≤1,(i=1,2,…,N
P).
式中,f
P(ξ)为0度方向方向性带隙目标函数;g
P为45度方向性通带约束函数。S6:利用乘子法将步骤S5中所提的全带隙或方向带隙的约束优化问题转化为无约束优化问题:
For loop=1,2,......
find ξ={ξ
1ξ
2…ξ
M}
T
式中,loop为子优化问题个数;f
obj(ξ)为所求子优化问题目标函数;f(ξ)为原优化问题目标函数;p
0为将有约束优化问题转化为无约束优化问题的罚函数乘子;Ω
k为子优化问题的可行域。
采用自适应设计空间调整策略,限制声学超材料单胞的材料布局单次优化更新的变化程度,基于序列Kriging代理模型优化算法求解步骤S6中的优化模型,直至优化问题收敛。
进一步的,所述的步骤述S3中,声学超材料单胞内材料分布与等效材料属性的表征与映射的具体步骤为:
S33:在单胞内选与有限单元中心相重合的N
P个材料场函数观察点,建立求各观察点的距离相关函数矩阵
求解广义特征值问题广义特征值方程Cψ
k=λ
kψ
k。其中λ
k和ψ
k分别表示第k阶特征值和特征向量。
进一步的,基于材料场级数展开描述的全带隙或方向性带隙非梯度声学超材料拓扑优化模型,具体步骤为:
S54:(1)若考虑全带隙拓扑优化问题,则建立拓扑优化问题模型:
s.t.:(K(k)-ω
2M)U=0,
ξ
TW
iξ≤1,(i=1,2,…,N
P).
(2)若考虑方向性波传导优化问题,建立局部带隙拓扑优化问题模型:
s.t.(K(k)-ω
2M)u=0
ξ
TW
iξ≤1,(i=1,2,…,N
P).。
本发明的有益效果为:本方法通过少量独立设计变量,实现复杂声学超材料单胞结构的拓扑表征与材料性能的映射,大大降低了声学超材料单胞拓扑优化计算量,可有效克服声学超材料设计中的局部最优解困难,适用于全带隙、方向性带隙等多种功能声学超材料优化设计问题,该方法不需要带隙特性的灵敏度信息,可实现任意阶带隙的声学超材料优化设计,并且可以直接扩展到其他复杂声学特性的超材料设计中,便于与各种有限元商业和自研软件进行对接
图1为本发明实施例提供的全带隙和方向性带隙材料微结构设计。(a)全带隙声学超材料微结构示意图;(b)由方向性带隙声学超材料组成的波导管示意图;(c)全带隙声学超材料色散曲线示意图;(d)方向性带隙声学超材料色散曲线示意图;
图2基于非梯度优化算法的声学超材料优化流程图;
图3(a)指定阶全带隙声学超材料最优拓扑构型;
图3(b)为指定频率(30kHz)下全带隙声学超材料结构声波传输振幅云图;
图3(c)为指定频率(50kHz)下全带隙声学超材料结构声波传输振幅云图;
图4(a)指定阶方向性带隙声学超材料最优拓扑构型;
图4(b)指定频率(20kHz)下方向性带隙声学超材料构成的波导管左侧入射声波传输振幅云图;
图4(c)指定频率(20kHz)下方向性带隙声学超材料构成的波导管右侧入射声波传输振幅云图;
图4(d)指定频率(33kHz)下方向性带隙声学超材料构成的波导管左侧入射声波传输振幅云图;
图4(e)指定频率(33kHz)下方向性带隙声学超材料构成的波导管右侧入射声波传输振幅云图。
以下结合技术方案和附图详细叙述本发明的具体实施例。
一种基于非梯度优化算法的声学超材料拓扑优化设计方法,该拓扑优化方法通过材料场级数展开策略,实现少量参数表征声学超材料单胞的任意拓扑,建立这些设计参数与材料性能插值模型及声学材料动力学性能的映射关系。通过求解广义特征值方程,得到声学超材料的色散曲线,建立全带隙和方向性带隙声学超材料拓扑优化模型,基于序列Kriging代理模型优化算法求解优化问题。优化过程避免了求解复杂声学超材料设计问题的灵敏度信息,并且能有效克服局部最优解陷阱。
图1描述的是全带隙和方向性带隙材料微结构设优化示意图。本发明以典型的铅—环氧树脂声学超材料单胞为例,其单胞尺寸为0.02m边长的正方形。选取第6-7阶全带隙和3-4阶段0度方向性带隙为优化问题的目标。图2是本方法的实施流程图。
S1:将声学材料单胞划分为1600x1600个平面应变单元,将时间域波传导 方程转化为基于有限元描述的频率域广义特征值方程(K(k)-ω
2M)U=0。
S3:对声学超材料单胞内材料分布与等效材料属性进行表征与映射。其具体步骤为:
S4:在简约布里渊区的ГXM边界等距选取0度,45度,90度波矢k各5组,通过求解任意可能的波矢k求解广义特征值问题(K(k)-ω
2M)U=0计算声学超材料的能带结构,得到前10阶频率对应的带隙ω
j(k
i)。
S5:以ξ={ξ
1ξ
2…ξ
M}
T为设计变量,考虑如下两种情况,分别建立声学超材料拓扑优化模型。
(51)指定第6阶和7阶的全带隙为目标,则拓扑优化目标函数可表达为 max:
引入材料场函数界限约束函数ξ
TW
iξ≤1,(i=1,2,…,1600)使得材料表征函数值介于
之间,进而建立拓扑优化问题模型(相关优化计算结果如图3所示):
s.t.:(K(k)-ω
2M)U=0,
ξ
TW
iξ≤1,(i=1,2,…,1600).
(52)指定第3阶和4阶的方向性带隙为目标,则选取ГX为带隙建立目标函数为max:
设置ГM方向通带约束
实现在45度方向上允许声波通过,引入材料场函数界限约束函数ξ
TW
iξ≤1,(i=1,2,…,1600)使得材料表征函数值介于
之间,进而建立拓扑优化问题模型(相关优化计算结果如图4所示):
s.t.(K(k)-ω
2M)u=0
ξ
TW
iξ≤1,(i=1,2,…,1600).
S6:利用乘子法将声学超材料带隙的约束优化问题转化为无约束优化问题:
For k=1,2,......
find ξ={ξ
1ξ
2…ξ
50}
T
采用自适应设计空间调整策略,限制声学超材料单胞的材料布局单次优化更新的变化程度,基于序列Kriging代理模型优化求解算法求解优化问题,直至优化问题收敛。
以6-7阶全带隙为目标的声学超材料非梯度拓扑优化方法所得声学超材料最优微结构如图3(a)。优化后全带隙声学超材料组成的宏观结构在30kHz和50kHz声波传输振幅云图见图3(b),结果表明全带隙声学超材料可阻断其带隙频率段内声波的传播,并允许不在其带隙内的声波通过。
以3-4阶方向性带隙为目标的声学超材料非梯度拓扑优化方法所得声学超材料最优微结构如图4(a)。由方向性声学超材料微结构组成的波导管结构在20kHz和33kHz下声波传输振幅云图见图4(b),结果表明方向性带隙声学超材料在带隙频率范围内能可阻断0度方向传播的声波而45度方向传播的声波可以自由通过,在带隙外各方向的声波均可通过。
本发明提供的能够有效克服声学超材料拓扑优化设计中的初始解依赖性和局部最优解陷阱难题,并能稳定高效的获得各阶带隙的声学超材料最优构型。
其对前述各实施例所记载的优化问题的带隙性能目标函数进行修改,或者对其中部分材料插值模型进行等同替换,并不使相应方法与方案的本质脱离本发明各实施例方法与方案的范围。
Claims (3)
- 一种基于非梯度拓扑优化的声学超材料设计方法,其特征在于,包括以下步骤:S1:将时间域声学超材料中弹性波的波传导问题转换为频率域下微结构的弹性波的广义特征值问题,即将时间域的波传导方程 转化为频率域的广义特征值方程(K(k)-ω 2M)U=0;采用平面应变单元对声学材料单胞进行离散,得到离散后的微结构总体刚度矩阵和总体质量矩阵;公式中字母指代为:r={x,y}表示位置向量;ρ(r)表示材料密度;U和 分别表示沿坐标轴x,y和z方向的位移和加速度向量;λ(r)和μ(r)表示随位置变化的拉梅系数;k表示平面波矢;K(k)表示与平面波矢相关的周期单胞总体刚度矩阵;M表示周期单胞的总体质量矩阵;ω表示圆频率;S2:基于Floquet–Bloch定理,在步骤S1中建立的广义特征值方程中引入周期性边界条件, 其中,U k(r)表示与位置向量r有相同周期性的周期位移场;基于有限元方法对微结构的特征值问题进行离散;S3:对步骤S2中离散的声学超材料单胞内材料分布与等效材料属性进行表征与映射;基于材料场级数展开策略,引入材料场函数 和距离相关函数C(r i,r j)描述单胞内部材料分布;利用级数展开降维技术,将声学超材料单胞内的材料分布用M个独立设计变量ξ={ξ 1 ξ 2 … ξ M} T进行表征;进而,引入RAMP材料插值模型,建立单元相对密度与材料等效杨氏模量的映射关系;S4:对步骤S3中声学超材料微结构的广义特征方程求解中,设置不同大小的平面波矢量k的入射波对简约布里渊区边界进行扫略,在简约布里渊区的ГXM边界等距选取0度、45度、90度波矢各5组,通过求解任意可能的波矢k求解广义特征值问题(K(k)-ω 2M)U=0;计算声学超材料的能带结构,并得到指定阶频率对应的带隙ω j(k i),式中,k i表示选取的第i个代表性平面波矢量,ω j(k i)表示在指定平面波矢量k i下单胞的第j阶特征圆频率;S5:根据步骤S4中带隙计算结果,以ξ={ξ 1 ξ 2 … ξ M} T为设计变量,建立全带隙或方向性带隙的声学超材料拓扑优化模型:(1)若指定第j阶和j+1阶的全带隙为目标,建立全带隙拓扑优化问题模型:s.t.(K(k)-ω 2M)u=0ξ TW iξ≤1,(i=1,2,…,N P).式中,f F(ξ)为全带隙目标函数;N P为总观察点个数;W i为与第i个观察点相关的特征矩阵;n k为简约布里渊区边界的计算点个数;(2)若指定第j阶和j+1阶的方向性带隙为目标,建立局部带隙拓扑优化问题模型:s.t.(K(k)-ω 2M)u=0ξ TW iξ≤1,(i=1,2,…,N P).式中,f P(ξ)为0度方向方向性带隙目标函数;g P为45度方向性通带约束函数;S6:利用乘子法将步骤S5中所提的全带隙或方向带隙的约束优化问题转化为无约束优化问题:For loop=1,2,......find ξ={ξ 1 ξ 2 … ξ M} T式中,loop为子优化问题个数;f obj(ξ)为所求子优化问题目标函数;f(ξ)为原优化问题目标函数;p 0为将有约束优化问题转化为无约束优化问题的罚函数乘子;Ω k为子优化问题的可行域;采用自适应设计空间调整策略,限制声学超材料单胞的材料布局单次优化更新的变化程度,基于序列Kriging代理模型优化算法求解步骤S6中的优化模型,直至优化问题收敛。
- 根据权利要求1所述的一种基于非梯度拓扑优化的声学超材料设计方法,其特征在于,所述的步骤述S3中,声学超材料单胞内材料分布与等效材料属性的表征与映射的具体步骤为:S33:在单胞内选与有限单元中心相重合的N P个材料场函数观察点,建立求各观察点的距离相关函数矩阵 求解广义特征值问题广义特征值方程Cψ k=λ kψ k;其中λ k和ψ k分别表示第k阶特征值和特征向量;
- 根据权利要求1或2所述的一种基于非梯度拓扑优化的声学超材料设计方法,其特征在于,基于材料场级数展开描述的全带隙或方向性带隙非梯度声学超材料拓扑优化模型,具体步骤为:S54:(1)若考虑全带隙拓扑优化问题,则建立拓扑优化问题模型:s.t.:(K(k)-ω 2M)U=0,ξ TW iξ≤1,(i=1,2,…,N P).(2)若考虑方向性波传导优化问题,建立局部带隙拓扑优化问题模型:
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